Kerr High-Precision Analysis (balanced)

Resolution: 1920×1080 | Inclination: 45.0° | Spin: 0.90 | Orders: 3 | r ∈ [2.3, 20.0]M

Hit Pixels
313027
15.1%
Miss Rate
84.9%
1760573 pixels
Order-1 Pixels
39035
Photon ring
NI p99
6.92e-12
Excellent

Hit / Miss & Orders

Distribution of geodesic orders per pixel. Multiple orders indicate strong gravitational lensing.

BucketPixelsShareVisual
Only order 0 (primary)27399213.2%
Orders 0 & 1 (+ ring)331481.6%
Orders ≥2 (+ subrings)58870.3%
No hit (miss)176057384.9%

Miss Taxonomy

Classification of rays that don't hit the accretion disc.

Miss Taxonomy 1760573 total misses 95.6% Escaped: 1682449 (95.6%)Captured: 78124 (4.4%)
Miss TypeCountShare
Escaped (r → ∞)168244995.6%
Captured (r → rh)781244.4%
Aborted (numerical)00.0%

Escaped: Ray reaches r > 1000M (escapes to infinity)
Captured: Ray falls into event horizon (r < rh + 0.01M)
Aborted: Integration stopped due to numerical issues or step limit

Order Mask Thumbnails

Visual spatial distribution of geodesic orders. White pixels indicate hits at that order.

Order 0 30.3% coverage

Primary disc image

Order 1 24.3% coverage

Photon ring

Order 2+ 11.5% coverage

Higher-order subrings

Outlier Spotlight

Top 10 rays with highest null invariant errors for quality inspection. These outliers warrant review but may still be within acceptable tolerances for complex geodesics.

RankPixel (x, y)OrderNI ErrorAttention Level
1(660, 671)08.40e-9High Attention
2(1060, 327)17.17e-9High Attention
3(1060, 327)27.17e-9High Attention
4(1275, 617)15.44e-9High Attention
5(1275, 617)25.44e-9High Attention
6(783, 608)04.77e-9High Attention
7(1274, 762)14.48e-9High Attention
8(841, 571)03.29e-9High Attention
9(1047, 435)13.26e-9High Attention
10(1028, 283)13.22e-9High Attention

Low: NI < 1e-12 | Medium: 1e-12 ≤ NI < 1e-9 | High Attention: NI ≥ 1e-9

Carter Constant Validation

The Carter constant K = Q + (Lz - aE)² must be ≥ 0 for physical geodesics. Violations indicate integration errors.

Carter Constant (K) Validation ✓ All geodesics physical (K ≥ 0) | 24485 blocks sampled | 0 negative Legend: K > 0 (physical) K ≈ 0 (roundoff) K < 0 (violation)

Transfer Functions

2D histograms showing how image position maps to disc emission radius for each geodesic order.

Transfer Function — Order 0 313027 rays ISCO Image Radius (px) Emission Radius (M) 0 454 20.0 2.3

Order 0 (Primary)

Transfer Function — Order 1 39035 rays ISCO Image Radius (px) Emission Radius (M) 0 453 20.0 2.3

Order 1 (Photon Ring)

Transfer Function — Order 2+ 5887 rays ISCO Image Radius (px) Emission Radius (M) 0 452 20.0 2.3

Order 2+ (Subrings)

Relative Light Travel Times

Heatmap showing arrival time delays across the image via affine parameter λ. Warm colors indicate longer light paths.

Relative Light Travel Time (Affine Parameter) Δλ: 54.2M — 182.2M (range: 128.0M) Color Scale: Earlier Later

Numerical Quality

Null invariant (NI) measures how well geodesics maintain the constraint gμν kμ kν = 0.

MetricValue
NI min0.00e0
NI median0.00e0
NI mean6.82e-13
NI p951.56e-12
NI p996.92e-12
NI max8.40e-9
NI < 1e-1561.0%
NI < 1e-1294.6%
NI < 1e-9100.0%

Good: Most rays are publishable quality, some numerical drift

Null Invariant Error Distribution (Log Scale) Null Invariant (NI) Count 01e-181e-151e-121e-91e-61e-3 1e-12 1e-9

Disc Interaction & Radial Distribution

Where disc hits occur. Peak near ISCO indicates brightest emission region.

Radial Hit Distribution by Order Radius (M) Hits 2.3 20.0 Order 0 Order 1 Order 2+
Radial Profile: Multi-Metric Analysis Radius (M) Normalized 2.3 20.0 ISCO Hit Count Redshift g φ-wraps

Binned Statistics (Order 0)

Radius RangeHitsShareVisual
[2.3-4.0M] Near ISCO75902.4%
[4.0-8.0M] Inner disc3345310.7%
[8.0-14.0M] Mid disc9794731.3%
[14.0-20.0M] Outer disc17403755.6%

Per-Order Statistics

φ-wraps are unitless (2π cycles), λ is affine parameter at disc hit

OrderCountMean φ-wrapsMean λ
0 (Primary)3130270.5963.6M
1 (Ring)390350.42102.5M
2+ (Subrings)58870.46140.8M

Redshift & Doppler Analysis

Redshift factor g = νobs/νemit (unitless). Values >1 indicate Doppler boosting (approaching), <1 indicate dimming (receding).

g min (most dimmed)0.884
g max (most boosted)5.009
g mean1.151
Asymmetry ratio2.19:1
Boosted (g>1)245617 hits (68.6%)
Dimmed (g<1)112332 hits (31.4%)

Clear Doppler asymmetry from frame-dragging

Angular Distribution (φ) Frame-dragging asymmetry 0° 90° 180° 270°

Light Bending & Lensing

φ-wraps measure how many times photons wind around the black hole before hitting the disc.

Orderφ-wraps RangeMean
0 (Primary)0.00 - 1.040.59
1 (Ring)0.42 wraps (avg)
2+ (Subrings)0.46 wraps (avg)

Max wraps: 1.04 (moderate lensing (typical for disc imaging))

Advanced Geodesic Diagnostics

Critical Curve & Shadow Fitting

Extracted 109172 boundary pixels between captured and escaped rays.

ParameterValue
Center (x, y)(895.8, 540.2)
Semi-major axis843.4 px
Semi-minor axis480.9 px
Rotation0.24°
Axis ratio (b/a)0.570

Fitted ellipse approximates the black hole shadow boundary. Axis ratio near 1.0 indicates circular shadow (low spin or face-on view).

Turning Points Distribution

Histogram of radial (r) and polar (θ) turning points. High counts indicate complex, chaotic geodesics near the photon sphere.

Turning Points Distribution Radial (r) 252669 105280 0 1 2 3 4 5 6 7 8 9 10+ Turning points Polar (θ) 263800 81589 11575 985 0 1 2 3 4 5 6 7 8 9 10+ Turning points

Wrap-Angle vs Impact Parameter

Relationship between impact parameter b = Lz/E and azimuthal wraps. The vertical red line marks the photon sphere radius where rays orbit multiple times before hitting the disc.

Wrap-Angle vs Impact Parameter b_photon ≈ 3.79 -17.9 -10.8 -3.6 3.5 10.7 17.8 0 0 1 1 2 2 Impact parameter b = L_z/E φ-wraps Order 0 Order 1 Order 2+
How to Read This Chart

X-axis (Impact Parameter b): Angular momentum per unit energy (Lz/E). Negative values mean counter-rotation relative to black hole spin.

Y-axis (φ-wraps): Number of times the light ray winds around the black hole before hitting the disc.

Colors: Green = Order 0 (primary), Blue = Order 1 (photon ring), Purple = Order 2+ (higher subrings)

Photon Sphere (red line): Critical impact parameter (~3.79M for a=0.9) where photons can orbit the black hole. Rays near this value show the highest wraps.

Expected Pattern: A peak in wraps near bphoton, with Order 1 and 2+ rays concentrated in this region due to multiple orbits before disc intersection.

Enhanced Outlier Detection

Outlier Summary

Comprehensive 2-pass outlier detection using robust statistical methods (MAD-based z-scores, Mahalanobis distance, spatial discontinuity analysis).

Critical
31
Physics violations
High
1329
Statistical anomalies
Medium
10580
Edge cases
Info
15289
Roundoff
Critical — 31 outliers (showing first 20 of 31)
Pixel Order Category Value Details
(560, 359) 0 Large Null Invariant Error 1.002e-9 NI=1.002e-9 (integration accuracy issue)
(660, 671) 0 Large Null Invariant Error 8.399e-9 NI=8.399e-9 (integration accuracy issue)
(703, 715) 0 Large Null Invariant Error 1.025e-9 NI=1.025e-9 (integration accuracy issue)
(763, 281) 0 Large Null Invariant Error 1.025e-9 NI=1.025e-9 (integration accuracy issue)
(770, 682) 0 Large Null Invariant Error 1.797e-9 NI=1.797e-9 (integration accuracy issue)
(780, 612) 0 Large Null Invariant Error 1.597e-9 NI=1.597e-9 (integration accuracy issue)
(783, 608) 0 Large Null Invariant Error 4.766e-9 NI=4.766e-9 (integration accuracy issue)
(819, 421) 0 Large Null Invariant Error 1.127e-9 NI=1.127e-9 (integration accuracy issue)
(819, 421) 1 Large Null Invariant Error 1.127e-9 NI=1.127e-9 (integration accuracy issue)
(841, 571) 0 Large Null Invariant Error 3.285e-9 NI=3.285e-9 (integration accuracy issue)
(847, 151) 0 Large Null Invariant Error 1.005e-9 NI=1.005e-9 (integration accuracy issue)
(870, 670) 0 Large Null Invariant Error 1.396e-9 NI=1.396e-9 (integration accuracy issue)
(879, 370) 0 Large Null Invariant Error 1.089e-9 NI=1.089e-9 (integration accuracy issue)
(879, 370) 1 Large Null Invariant Error 1.089e-9 NI=1.089e-9 (integration accuracy issue)
(941, 913) 0 Large Null Invariant Error 1.062e-9 NI=1.062e-9 (integration accuracy issue)
(971, 857) 0 Large Null Invariant Error 1.062e-9 NI=1.062e-9 (integration accuracy issue)
(971, 857) 1 Large Null Invariant Error 1.062e-9 NI=1.062e-9 (integration accuracy issue)
(1000, 917) 0 Large Null Invariant Error 1.076e-9 NI=1.076e-9 (integration accuracy issue)
(1000, 917) 1 Large Null Invariant Error 1.076e-9 NI=1.076e-9 (integration accuracy issue)
(1025, 241) 1 Large Null Invariant Error 1.329e-9 NI=1.329e-9 (integration accuracy issue)
High — 1329 outliers (showing first 20 of 1329)
Pixel Order Category Value Details
(523, 609) 0 Spatial Discontinuity 9.530e0 NI differs from neighbors (local MAD z=9.5, Δ=9.9%)
(524, 484) 0 Spatial Discontinuity 1.642e1 NI differs from neighbors (local MAD z=16.4, Δ=10.1%)
(524, 512) 0 Spatial Discontinuity 9.668e0 NI differs from neighbors (local MAD z=9.7, Δ=7.8%)
(526, 639) 0 Spatial Discontinuity 6.679e0 NI differs from neighbors (local MAD z=6.7, Δ=8.9%)
(532, 595) 0 Spatial Discontinuity 6.357e0 NI differs from neighbors (local MAD z=6.4, Δ=13.7%)
(538, 388) 0 Spatial Discontinuity 1.450e1 NI differs from neighbors (local MAD z=14.5, Δ=10.8%)
(538, 414) 0 Spatial Discontinuity 2.762e1 NI differs from neighbors (local MAD z=27.6, Δ=15.9%)
(539, 414) 0 Spatial Discontinuity 1.897e1 NI differs from neighbors (local MAD z=19.0, Δ=11.0%)
(540, 413) 0 Spatial Discontinuity 7.675e0 NI differs from neighbors (local MAD z=7.7, Δ=20.1%)
(544, 419) 0 Spatial Discontinuity 5.646e1 NI differs from neighbors (local MAD z=56.5, Δ=25.1%)
(551, 372) 0 Spatial Discontinuity 9.645e0 NI differs from neighbors (local MAD z=9.6, Δ=11.6%)
(551, 616) 0 Spatial Discontinuity 1.220e1 NI differs from neighbors (local MAD z=12.2, Δ=9.5%)
(552, 707) 0 Spatial Discontinuity 6.302e0 NI differs from neighbors (local MAD z=6.3, Δ=19.5%)
(554, 730) 0 Spatial Discontinuity 6.416e0 NI differs from neighbors (local MAD z=6.4, Δ=5.6%)
(557, 533) 0 Spatial Discontinuity 1.215e1 NI differs from neighbors (local MAD z=12.2, Δ=16.7%)
(560, 460) 0 Spatial Discontinuity 1.378e1 NI differs from neighbors (local MAD z=13.8, Δ=13.0%)
(563, 358) 0 Spatial Discontinuity 2.249e1 NI differs from neighbors (local MAD z=22.5, Δ=10.5%)
(563, 703) 0 Spatial Discontinuity 1.064e2 NI differs from neighbors (local MAD z=106.4, Δ=12.1%)
(564, 670) 0 Spatial Discontinuity 8.512e0 NI differs from neighbors (local MAD z=8.5, Δ=11.7%)
(577, 720) 0 Spatial Discontinuity 7.355e1 NI differs from neighbors (local MAD z=73.5, Δ=11.8%)
Medium — 10580 outliers (showing first 20 of 10580)
Pixel Order Category Value Details
(640, 544) 0 Extreme φ-Wraps 9.584e-1 φ-wraps=0.96 > p99=0.96
(640, 547) 0 Extreme φ-Wraps 9.582e-1 φ-wraps=0.96 > p99=0.96
(643, 563) 0 Extreme φ-Wraps 9.583e-1 φ-wraps=0.96 > p99=0.96
(643, 567) 0 Extreme φ-Wraps 9.577e-1 φ-wraps=0.96 > p99=0.96
(643, 571) 0 Extreme φ-Wraps 9.571e-1 φ-wraps=0.96 > p99=0.96
(644, 541) 0 Extreme φ-Wraps 9.609e-1 φ-wraps=0.96 > p99=0.96
(645, 541) 0 Extreme φ-Wraps 9.615e-1 φ-wraps=0.96 > p99=0.96
(645, 544) 0 Extreme φ-Wraps 9.613e-1 φ-wraps=0.96 > p99=0.96
(645, 571) 0 Extreme φ-Wraps 9.583e-1 φ-wraps=0.96 > p99=0.96
(646, 560) 0 Extreme φ-Wraps 9.604e-1 φ-wraps=0.96 > p99=0.96
(646, 577) 0 Extreme φ-Wraps 9.578e-1 φ-wraps=0.96 > p99=0.96
(648, 566) 0 Extreme φ-Wraps 9.608e-1 φ-wraps=0.96 > p99=0.96
(650, 548) 0 Extreme φ-Wraps 9.640e-1 φ-wraps=0.96 > p99=0.96
(650, 552) 0 Extreme φ-Wraps 9.637e-1 φ-wraps=0.96 > p99=0.96
(650, 554) 0 Extreme φ-Wraps 9.635e-1 φ-wraps=0.96 > p99=0.96
(650, 581) 0 Extreme φ-Wraps 9.593e-1 φ-wraps=0.96 > p99=0.96
(650, 593) 0 Extreme φ-Wraps 9.567e-1 φ-wraps=0.96 > p99=0.96
(651, 546) 0 Extreme φ-Wraps 9.648e-1 φ-wraps=0.96 > p99=0.96
(651, 555) 0 Extreme φ-Wraps 9.640e-1 φ-wraps=0.96 > p99=0.96
(651, 569) 0 Extreme φ-Wraps 9.621e-1 φ-wraps=0.96 > p99=0.96
Info — 15289 outliers (showing first 20 of 15289)
Pixel Order Category Value Details
(509, 523) 0 Long Affine Parameter (Outer Disc) 8.391e1 λ=83.9M (MAD z=4.2, outer disc edge)
(509, 530) 0 Long Affine Parameter (Outer Disc) 8.391e1 λ=83.9M (MAD z=4.2, outer disc edge)
(509, 532) 0 Long Affine Parameter (Outer Disc) 8.391e1 λ=83.9M (MAD z=4.2, outer disc edge)
(509, 543) 0 Long Affine Parameter (Outer Disc) 8.391e1 λ=83.9M (MAD z=4.2, outer disc edge)
(509, 544) 0 Long Affine Parameter (Outer Disc) 8.391e1 λ=83.9M (MAD z=4.2, outer disc edge)
(509, 552) 0 Long Affine Parameter (Outer Disc) 8.392e1 λ=83.9M (MAD z=4.2, outer disc edge)
(509, 556) 0 Long Affine Parameter (Outer Disc) 8.392e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 502) 0 Long Affine Parameter (Outer Disc) 8.388e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 508) 0 Long Affine Parameter (Outer Disc) 8.387e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 518) 0 Long Affine Parameter (Outer Disc) 8.386e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 528) 0 Long Affine Parameter (Outer Disc) 8.386e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 545) 0 Long Affine Parameter (Outer Disc) 8.386e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 553) 0 Long Affine Parameter (Outer Disc) 8.387e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 558) 0 Long Affine Parameter (Outer Disc) 8.388e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 560) 0 Long Affine Parameter (Outer Disc) 8.388e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 565) 0 Long Affine Parameter (Outer Disc) 8.389e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 566) 0 Long Affine Parameter (Outer Disc) 8.389e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 568) 0 Long Affine Parameter (Outer Disc) 8.390e1 λ=83.9M (MAD z=4.2, outer disc edge)
(510, 570) 0 Long Affine Parameter (Outer Disc) 8.390e1 λ=83.9M (MAD z=4.2, outer disc edge)
(511, 496) 0 Long Affine Parameter (Outer Disc) 8.385e1 λ=83.8M (MAD z=4.2, outer disc edge)

Spatial Outlier Distribution

Downsampled heatmap showing where outliers are concentrated. Darker = more outliers in region, color = highest severity level.

Outlier Spatial Distribution27229 outliers detected across 3886 regionsSeverity:CriticalHighMediumInfoDensity:darker = more outliers
About This Analysis

This report analyzes high-precision f64 geodesic data from Kerr spacetime ray tracing. Each pixel traces a null geodesic (light path) backward from the camera through curved spacetime, potentially crossing the accretion disc multiple times due to extreme gravitational lensing.

The null invariant (NI) measures numerical accuracy. Values <1e-9 indicate excellent geodesic integration.